Requirements

A transmission line requires two conductors as sketched in fig. 1, which shows a 2-wire line such a telephone line, a coaxial cable as found in cable TV distribution, and a microstrip line as found in cell phone RF interconnects.

A two-wire line becomes a transmission line when the wavelength of operation becomes comparable to the size of the line (or higher spectral component for pulses). In general a transmission line much support (TEM) transverse electromagnetic modes.

Time harmonic solutions on transmission lines

In fig. 2, an electronic representation of a transmission line circuit is sketched.

Note that \( \eta = \sqrt{\mu_0/\epsilon_0} = 120 \pi \Omega \) is the intrinsic impedance of free space. The values \( a, b \) in \ref{eqn:uwaves4TransmissionLines:500} can be used to tune the characteristic impedance of the transmission line.

Lossless line.

The lossless lossless case where \( R = G = 0 \) was considered above. The results were

The shape is preserved but the amplitude has an exponential attenuation along the line.

In this case, since \( \beta(\omega) \) is a linear function to first order, we have no dispersion. All of the Fourier components of a pulse travel with the same phase velocity since \( v_\phi = \omega/\beta \) is constant. i.e. \( v(z, t) = e^{-\alpha z} f( t – z/v_\phi ) \). We should expect dispersion when the \( R/\omega L \) and \( G/\omega C \) start becoming more significant.

Distortionless line.

Motivated by the early telegraphy days, when low loss materials were not available. Therefore lines with a constant attenuation and constant phase velocity (i.e. no dispersion) were required in order to eliminate distortion of the signals. This can be achieved by setting

This difference is the power delivered to the load. This is not z-dependent because we are considering the lossless case. Maximum power is delivered to the load when \( \Gamma_{\textrm{L}} = 0 \), which occurs when the impedances are matched.

In our final problem set we used the impedance transformation for calculations related to a microslot antenna. This transformation wasn’t familiar to me, and is apparently covered in the third year ECE fields class. I found a derivation of this in [1], but the idea is really simple and follows from the reflection coefficient calculation for a normal reflection configuration.

Consider a normal field reflection between two interfaces, as sketched in fig. 1.

The field orientations have been picked so that the tangential component of the electric field is \( \xcap \) oriented for all of the incident, reflected, and transmitted components. Requiring equality of the tangential field components at the interface gives